Graph (mathematics)

This article just presents the basic definitions. For a broader view see graph theory. For another mathematical use of "graph", see graph of a function.

In mathematics and computer science a graph is the basic object of study in graph theory. Informally, a graph is a set of objects called vertices joined by links called edges. Typically, a graph is depicted as a set of dots (vertices, nodes) joined by lines (the edges). Depending on the application some edges can be directed.

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Definitions

Definitions in graph theory vary in the literature. Here are the conventions used in this encyclopedia.

Undirected graph

An undirected graph or graph G is an ordered pair G:=(V, E) with

• V a set of vertices or nodes,
• E a set of unordered pairs of distinct vertices, called edges or lines.

V (and hence E) are usually taken to be finite sets, and many of the well-known results are not true (or are rather different) for infinite graphs because many of the arguments fail in the infinite case.

Directed graph

A directed graph, digraph or quiver G is an ordered pair G:=(V,A) with

• V a set of vertices or nodes,
• A a set of ordered pairs of vertices called directed edges, arcs or arrows.

A variation on this definition is the oriented graph, which is a simple graph with an orientation assigned to all of its edges. The distinction between a directed graph and an oriented graph is that if x and y are vertices, a directed graph allows both (x, y) and (y, x) as edges, while only one is permitted in an oriented graph.

A directed graph may or may not be allowed to have loops, that is, edges where the start and end vertices are the same. By definition, this is forbidden in an oriented graph.

Mixed graph

A mixed graph G is a 3-tuple G:=(V,E,A) with V, E and A defined as above.

Variations in the definitions

As defined above, edges of undirected graphs have distinct ends, and E and A are sets (with distinct elements as sets always do). Many applications require more general possibilities, but terminology varies. A loop is an edge (directed or undirected) with both ends the same; these may be permitted or not permitted according to the application. Sometimes E or A are allowed to be multisets, so that there can be more than one edge between the same two vertices. The unqualified word "graph" might allow or disallow multiple edges in the literature, according to the preferences of the author. If it is intended to exclude multiple edges (and, in the undirected case, to exclude loops), the graph can be called simple. On the other hand, if it is intended to allow multiple edges, the graph can be called a multigraph. Sometimes the word pseudograph is used to indicate that both multiple edges and loops are allowed.

Further definitions

For more definitions see Glossary of graph theory.

Two edges of a graph are called adjacent (sometimes coincident) if they share a common vertex. Similarly, two vertices are called adjacent if they share a common edge, that is they are joined by an edge. An edge and a vertex on that edge are called incident.

The graph with only one vertex and no edges is the trivial graph. A graph with only vertices and no edges is known as an empty graph; the graph with no vertices and no edges is the null graph, but not all mathematicians allow this concept.

In a weighted graph or digraph, each edge is associated with some value, variously called its cost, weight, length or other term depending on the application; such graphs arise in many contexts, for example in optimal route problems such as the traveling salesman problem.

Normally, the vertices of a graph by their nature are indistinguishable. (Of course, they may be distinguishable by the properties of the graph itself, e.g., by the numbers of incident edges). Some branches of graph theory require to uniquely identify vertices. If each vertex is given a label, then the graph is said to be a vertex-labeled graph, whereas graphs which have labeled edges are called edge-labeled graphs. Graphs with labels attached to edges or vertices are more generally designated as labeled. Consequently, graphs without labels are called unlabelled.

Examples The picture is a graphic representation of the following graph

• V:={1,2,3,4,5,6}
• E:={{1,2},{1,5},{2,3},{2,5},{3,4},{4,5},{4,6}}

Generalizations

In a hypergraph, an edge can join more than two vertices.

An undirected graph can be seen as a simplicial complex consisting of 1-simplices (the edges) and 0-simplices (the vertices). As such, complexes are generalizations of graphs since they allow for higher-dimensional simplices.

Every graph gives rise to a matroid, but in general the graph cannot be recovered from its matroid, so matroids are not truly generalizations of graphs.

In model theory, a graph is just a structure. But in that case, there is no limitation on the number of edges: it can be any cardinal number.

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